\section{Samples} 
\label{sec:samples}

The samples used in this analysis come from AMPT versioin 1.26t1b. The collision frame is Au-Au. The impact parameter in this research  is set as 8 fm. The parton screening mass is 3.2264 $fm^{-1}$. $\alpha$ sets in parton cascade is 0.33. 
In this analysis, we generated about 600,000 events and we focused on the parton level transformation of these samples. In the output files, there are two files about the parton level information. 
The first one is "parton-initial-afterPropagation.dat". In this file, all the parton initial state are recorded. We can find momentum, position, mass information in it. 
The other file is "parton-collisionsHistory.dat". This file is used to store the information of collisions. Each collision is recorded with four lines. The first two lines are the two partons before the collision and last two lines are these two partons after the collision.
In other to get the collision-chain, we compare the final state informaiton and initial state information between collisions. 
Based on the parton momentum , position and PDGId, we trace the parton from initial status to make a "collision chain" to final parton, which parton will not collide anymore. 
After getting the chain of a parton collision, we can know the number of collision of each parton. And all the information before the parton collide on other partons ( we call it initial parton) and this parton after all collision( final parton). So that we can also research on the intermediate state of the partons.

After we get the collision chain, we can calculate the number of collisions. For a certain parton, we can both get the number of collisions from its initial state to final state, and  we can also find the number of collision in any of the parton's intermediate state. 
We define the total number of collisions as $N_{Coll}$ and number of collision in mediate state is $N^{*}_{Coll}$. Both of them are very meaningful in the following research.


Because the heavy-ion collisions are not always central, there will be an overlap region of the colliding nuclei.
In transverse plane the shape of this region is like ellipse. In the process of the scatterings, the spatial anisotropy will transform from initial state into final state.
In order to quantify the momentum space anisotropy, we can expand the azimuthal distribution of the partons in the Fourier decompositions.
The second Fourier coefficient is the largest contribution of shape of plane, we define it as $v_{2}$, the elliptic flow coefficient.
It is very important to study the how the ellipse transform in collisions. 
As the definition of $v_{2}$ we can can calculate the average value of $cos(2*\phi$ to study the relationship between $v_{2}$ and number of collisions or transverse momentum.

